Development and analysis of an adaptive space–time finite element method for nonlinear parabolic equations of p-Laplace type

The Oberwolfach mini-workshop “Convergence of Adaptive Algorithms” originated from a previous Oberwolfach meeting 16/2004 on the topic of “Self-adaptive Methods for Partial Differential Equations” which took place in Spring 2004. One motivation for the mini-workshop was the resolution of the key issues of ‘error reduction’ in adaptive finite element schemes and the necessity, or otherwise, for ‘coarsening strategies’ in adaptive algorithms. While the former topic might be regarded as more theoretical, the latter has important practical repercussions in the sense that essentially every practical numerical example would indicate that coarsening is unnecessary. However, the existing proofs of optimal complexity would seem to suggest that coarsening is essential if one is to control discretisation error at an optimal computational cost. Set against this background, the mini-workshop comprised of 18 leading experts on the convergence of adaptive finite element methods representing 8 different countries and three continents, who identified and discussed the following specific open questions: During the mini-workshop 11 talks were given concerning adaptive finite element methods and covering a range of new extensions to the classical convergence analysis were presented. The talks directly addressed the important issues including the role of coarsening, marking rules, hp -adaptive refinement strategies, discrete weighted residual (DWR) adaptive methods in addition to the convergence of non-conforming and mixed methods. The participants also presented very recent work on applications to new classes of equations, e.g. for rough and non-conforming obstacles, for the Laplace–Beltrami operator and the Stokes equations. The presentations were complemented by several more wide-ranging discussion sessions on open questions and future directions in the field. In particular, it was widely felt that in the case of the class of adaptive algorithms for which there is a proof of optimality, more numerical experiments are necessary to achieve a deeper understanding of the insights and issues highlighted by the abstract analysis. Moreover, numerical experiments were seen to be important in providing quantitative information on the generic constants that appear in the abstract error bounds, where it appears infeasible to derive realistic estimates of the constants that arise in the existing theory. In a similar vein, it would also be of considerable interest to quantify the saving in computational effort through the use of different adaptive schemes and in comparison to uniform refinement. Furthermore, the theory may be used to identify specific examples where coarsening steps are really needed to attain an optimal algorithm. More generally, the identification of a suite benchmark tests and comparisons with other adaptive strategies, for which current theory is lacking, was also suggested. The importance of understanding the relationship between the numerical solution and the best approximation in the pre-asymptotic range as one can construct problems for which the cost of computations in the asymptotic range is prohibitively high. Duality-based adaptive strategies compute a weighting of the relevance of the data in the course of the calculation. Starting with this aspect, it was also discussed how the convergence analysis of adaptive algorithms can be related to a data analysis of the problem. Participants proposed that the analysis of duality-based strategies provides an indication that after sufficiently many adaptive refinement steps it may simply be the case that the best strategy to continue the computation with uniform refinement. The presence of singularities in the solution may play a subtle role here. Part of the session was dedicated to adaptive refinement strategies in three space dimensions, including the question of convergence of adaptive methods in this setting. Another topic hotly discussed were outstanding hp -approximation issues. Participants agreed that automatic decision mechanisms when h - and when p -refinement is preferable but that there is a definite need for further fundamental improvements. The issue of the development and analysis of reliable and efficient error estimators is less developed for the p - and hp -version of the finite element method than for the h -version. Similarly, convergence proofs for hp -adaptive finite element methods need to be addressed in future in more detail.

A goal-oriented adaptive frequency domain finite element method for solving electromagnetic radiation problems including complex structures is presented in this paper. Compared with the traditional adaptive finite element method, the goal-oriented method can flexibly control the refined regions according to the parameters of interest; therefore it has better convergence and has made significant progress in scattering problems and eigenvalue problems. To simulate complex antennas, this paper proposes an error indicator with high accuracy and low computational cost, and it uses the adjoint problem to weight element residuals without additional degrees of freedom. Moreover, high-quality mesh refinement algorithms adapted to this indicator are developed using a suitable point insertion strategy for multiscale structures. By simulating two practical antennas, comparisons with the traditional goal-oriented FEM and the well-developed <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">h</i> -adaptive finite element method in commercial software demonstrate the accuracy and efficiency of the proposed method.

In this paper we start a systematic investigation of applying an adaptive finite element method to the Einstein equations, especially binary compact object simulations. To our knowledge, this is the first study on this topic. Puncture type initial data are solved with the adaptive finite element method. The numerical scheme proposed in the current work can be straightforwardly extended to the general type of initial data of the Einstein equations. The Parallel Hierarchical Grid library and the existing numerical relativity code AMSS-NCKU are used to develop the adaptive finite element Einstein solver. In the unsmooth toy model problem, the adaptive mesh refinement operation can catch the unsmooth region efficiently. The numerical solution deviates the exact solution by an error less than $1{0}^{\ensuremath{-}5}$. In the binary black hole problem, our solution is consistent with the one gotten by the TwoPuncture code which uses a pseudospectral method. As we expected, the solution gotten by the finite element method is less accurate than that gotten by the spectral method. But the relative error distributes almost uniformly. The adaptive mesh refinement method is quite efficient and it does not waste computational effort. Our finite element code is more flexible than the TwoPuncture code. It can be used to treat other general initial data problems such as the three black holes problem, besides the binary black hole problem. We test one typical three black holes problem also. In all of the test cases, our adaptive finite element code works quite well.

The need to develop reliable and efficient adaptive algorithms using mixed finite element methods arises from various applications in fluid dynamics and computational continuum mechanics. In order to save degrees of freedom, not all but just some selected set of finite element domains are refined and hence the fundamental question of convergence requires a new mathematical argument as well as the question of optimality.We will present a new adaptive algorithm for mixed finite element methods to solve the model Poisson problem, for which optimal convergence can be proved. The a posteriori error control of mixed finite element methods dates back to Alonso (1996) Error estimators for a mixed method. and Carstensen (1997) A posteriori error estimate for the mixed finite element method. The error reduction and convergence for adaptive mixed finite element methods has already been proven by Carstensen and Hoppe (2006) Error Reduction and Convergence for an Adaptive Mixed Finite Element Method, Convergence analysis of an adaptive nonconforming finite element methods.Recently, Chen, Holst and Xu (2008) Convergence and Optimality of Adaptive Mixed Finite Element Methods. presented convergence and optimality for adaptive mixed finite element methods following arguments of Rob Stevenson for the conforming finite element method. Their algorithm reduces oscillations, before applying and a standard adaptive algorithm based on usual error estimation. The proposed algorithm does this in a natural way, by switching between the reduction of either the estimated error or oscillations. (© 2008 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Error reduction, convergence and optimality are analyzed for adaptive mixed finite element methods (AMFEM) for diffusion equations without marking the oscillation of data. Firstly, the quasi-error, i.e. the sum of the stress variable error and the scaled error estimator, is shown to reduce with a fixed factor between two successive adaptive loops, up to an oscillation. Secondly, the convergence of AMFEM is obtained with respect to the quasi-error plus the divergence of the flux error. Finally, the quasi-optimal convergence rate is established for the total error, i.e. the stress variable error plus the data oscillation.

Convergence of a standard adaptive nonconforming finite element method with optimal complexity

In this paper we present a way to numerically simulate large deformations of incompressible material and the corresponding hydrostatic pressure by using a mixed, adaptive finite element method (FEM). Starting from the system of differential equations we will derive the weak nonlinear formulation, which can be solved with a Newton's method. In every iteration step this will lead to a saddle point problem. By using Taylor Hood finite elements we will obtain a discrete, indefinite problem, which can be handled with the Bramble Pasciak conjugate gradient method. Finally we give a numerical example. (© 2011 Wiley‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Adaptive nearest-nodes finite element method guided by gradient of linear strain energy density

Convergence analysis of an adaptive continuous interior penalty finite element method for the Helmholtz equation

We consider the design of an effective and reliable adaptive finite element method (AFEM) for the nonlinear Poisson-Boltzmann equation (PBE). We first examine the two-term regularization technique for the continuous problem recently proposed by Chen, Holst, and Xu based on the removal of the singular electrostatic potential inside biomolecules; this technique made possible the development of the first complete solution and approximation theory for the Poisson-Boltzmann equation, the first provably convergent discretization, and also allowed for the development of a provably convergent AFEM. However, in practical implementation, this two-term regularization exhibits numerical instability. Therefore, we examine a variation of this regularization technique which can be shown to be less susceptible to such instability. We establish a priori estimates and other basic results for the continuous regularized problem, as well as for Galerkin finite element approximations. We show that the new approach produces regularized continuous and discrete problems with the same mathematical advantages of the original regularization. We then design an AFEM scheme for the new regularized problem, and show that the resulting AFEM scheme is accurate and reliable, by proving a contraction result for the error. This result, which is one of the first results of this type for nonlinear elliptic problems, is based on using continuous and discrete a priori L(∞) estimates to establish quasi-orthogonality. To provide a high-quality geometric model as input to the AFEM algorithm, we also describe a class of feature-preserving adaptive mesh generation algorithms designed specifically for constructing meshes of biomolecular structures, based on the intrinsic local structure tensor of the molecular surface. All of the algorithms described in the article are implemented in the Finite Element Toolkit (FETK), developed and maintained at UCSD. The stability advantages of the new regularization scheme are demonstrated with FETK through comparisons with the original regularization approach for a model problem. The convergence and accuracy of the overall AFEM algorithm is also illustrated by numerical approximation of electrostatic solvation energy for an insulin protein.

An adaptive enriched semi-Lagrangian finite element method for coupled flow-transport problems

The convergence analysis with rates for adaptive least-squares finite element methods (ALSFEMs) combines arguments from the a posteriori analysis of conforming and mixed finite element schemes. This paper provides an overview of the key arguments for the verification of the axioms of adaptivity for an ALSFEM for the solution of a linear model problem. The formulation at hand allows for the simultaneous analysis of first-order systems of the Poisson model problem, the Stokes equations, and the linear elasticity equations. Following [Carstensen and Park, SIAM J. Numer. Anal. 53 (1), 2015], the adaptive algorithm is driven by an alternative residual-based error estimator with exact solve and includes a separate marking strategy for quasi-optimal data resolution of the right-hand side. This presentation covers conforming discretisations for an arbitrary polynomial degree and mixed homogeneous boundary conditions.

Development of automatic shape design optimization algorithms requires the consideration of convergence issues. One issue is the approximation accuracy of the objective function and its gradient. This is of particular concern in shape optimization problems, where shape changes during the design iterations may drastically effect the flow solution. Thus, the discretization selected for the initial design guess may not be appropriate as the design evolves. It is natural to consider adaptive solvers for these problems. In this work, we consider the use of an adaptive finite element solver in computing flow variables and their sensitivities for incompressible, viscous flows. Introduction and Motivation The mathematical description of optimal design problems requires minimizing a design objective function over a set of admissible designs. In physical systems, the design often satisfies a constraint which is modeled as the solution of a partial differential equation (with appropriate boundary conditions) in the parameter dependent domain. We consider problems where the constraints are given by the two-dimensional Navier-Stokes equations. For most practical problems, the solution to the partial differential equation, and hence the solution to the optimal design problem, needs to be approximated. A survey of many approximation methods can be found in Frank and Shubin [11], We focus on implementation issues for a black-box method. This method uses an approximate-then-optimize approach, where the design objective function is first approximated, then an optimization algorithm is used to find parameters which minimize this approximate objective function. The convergence and efficiency of this method depend on the optimization algorithm used and the approximations used for the objective function (and its gradient). We consider the case where the Navier-Stokes equations are approximated by a finite element method. As the parameters are updated by the optimization algorithm, the original discretization may not be appropriate. This is particularly true when there are large changes in the shape of the domain. Since an automatic design program is desired, where there is no need for human intervention at the intermediate designs, it is natural to consider adaptive mesh refinement techniques in the approximations. These techniques change the discretization according to a measure of the error in the flow approximation. Since the exact flow solution is not known a priori, this error function is estimated using the computed solution [21]. There are three strategies for adaptivity, p-, r-, and h-refmement (and combinations of them). The prefinement relies on increasing the order of those finite elements where the error estimator is predicted to be large. This refinement scheme was considered in the context of structural shape optimization by [18] and [20]. This type of strategy does not readily apply to a mixed finite element formulation, such as that used to approximate the Navier-Stokes equations, due to the need to satisfy the LBB condition [7]. The r-refinement strategy consists of relocating the node points of the elements in order to uniformly distribute the error over all of the elements [14]. The h-refinement strategy subdivides elements in areas where the error is large. It has been Copjrijhl ©I9» by the «mhon. PublMnd by U* American Inslilule of Aeranjouiat and Astronxutlcs. Inc. with permiuioa. 1 American Institute of Aeronautics and Astronautics shown [21] that as the mesh is uniformly refined, that the error estimator will vanish. An h-remeshing adaptive finite element method has been demonstrated for solving complex flow problems, see [15], [16] and the references therein. This type of refinement scheme has been applied to structural shape optimization problems by [1], [8], [10] and [13]. When using a gradient-based optimization algorithm in the black-box method, the gradient needs to be computed efficiently in order to make the method practical. One technique is to use flow sensitivities, the derivative of the flow variables with respect to the design parameters. These sensitivities are commonly computed by differentiating the algorithm used to approximate the flow variables. Programs such as ADDFOR [2] can perform this differentiation automatically. However, since the discretization is parameter dependent, the derivatives of the discretization with respect to the design parameters (mesh sensitivities) need to be determined. For an adaptive solution scheme, these mesh sensitivities are not readily available, however an ad-hoc approach is demonstrated in [8]. We consider an alternate approach based on approximating the sensitivity equation, obtained by differentiating the partial differential equation and its boundary conditions with respect to the design parameters [4]. This equation is approximated efficiently using the same approximation scheme used to approximate the flow. Moreover, since the differentiation is performed before the approximation, there is no need to compute mesh sensitivities. Although the operations of approximation and differentiation don't commute in general, the convergence of the optimization algorithm using these computed sensitivities has been addressed using the notion of asymptotic consistency [3], [5]. In fact, this sensitivity equation technique has been successfully used in conjunction with a finite element approximation of the Navier-Stokes equations for a shape optimization problem [9]. This paper is organized as follows. In the next section, we introduce the sensitivity equations for the NavierStokes equations and the finite element approximation scheme. We then introduce the adaptive refinement strategy. We present two numerical examples, the first is a cylinder submerged in a channel where we demonstrate the accuracy of the sensitivity calculations and the second is a 2D shear layer example which motivates future work. In the following section, we provide our conclusions. Navier-Stokes Sensitivity Equations Navier-Stokes Equations In this paper, we consider optimal design problems where the constraint is described by the two-dimensional Navier-Stokes equations, V a = 0, (1) p (u • Vu) V • r(n) + V/» = F, (2) where p is the (constant) density, = («, v) is the velocity, P is the pressure, r is the stress tensor defined by with (constant) viscosity fi and F is the external body force. The solution of (l)-(2) satisfies the appropriate Dirichlet or Neuman boundary conditions in the (possibly parameter dependent) domain £2. The finite element equations for the flow are obtained by writing equations (l)-(2) in weak form, i.e. (V • u, u;) = 0 (pu Vu, v) + a(n, v) (P, V • v) = (F, v) , where (-, •) is the usual L inner product, (u, v) = / • v dfi Jn and a(-, •) is the bilinear form fl(u, v) = / r(u) : Vv di2. JQ These equations are solved in primitive variables using an augmented Lagrangian technique to treat the incompressibility. Discretization is performed with seven node Crouzeix-Raviart triangular elements (which use an enriched quadratic velocity approximation and a discontinuous linear pressure) [7]. The result is a set of nonlinear algebraic equations for the flow. Using Newton's method to solve these equations results in solving a linear system of the form 1 I A + -BM-B -(L(u) 1 AU = [AH + BM-'Bu + N(u)u f] (3) for the update Au, where the matrices A, B, L, M and N are formed using the current flow solution u. The flow at the next iteration is then obtained using the update = u -|Au*. (4) American Institute of Aeronautics and Astronautics This process is repeated until the residual of the nonlinear equations satisfies a prescribed tolerance. We denote the final converged solution by p (s • Vu) + p (u • Vs)

We are concerned with the numerical solution of distributed optimal control problems for second order elliptic variational inequalities by adaptive finite element methods. Both the continuous problem as well as its finite element approximations represent subclasses of Mathematical Programs with Equilibrium Constraints (MPECs) for which the optimality conditions are stated by means of stationarity concepts in function space (Hintermüller and Kopacka, SIAM J. Optim. 20:868–902, 2009) and in a discrete, finite dimensional setting (Scheel and Scholtes, Math. Oper. Res. 25:1–22, 2000) such as (\(\varepsilon\)-almost, almost) C- and S-stationarity. With regard to adaptive mesh refinement, in contrast to the work in (Hintermüller, ESAIM Control Optim. Calc. Var., 2012, submitted) which adopts a goal oriented dual weighted approach, we consider standard residual-type a posteriori error estimators. The first main result states that for a sequence of discrete C-stationary points there exists a subsequence converging to an almost C-stationary point, provided the associated sequence of nested finite element spaces is limit dense in its continuous counterpart. As the second main result, we prove the reliability and efficiency of the residual-type a posteriori error estimators. Particular emphasis is put on the approximation of the reliability and efficiency related consistency errors by heuristically motivated computable quantities and on the approximation of the continuous active, strongly active, and inactive sets by their discrete counterparts. A detailed documentation of numerical results for two representative test examples illustrates the performance of the adaptive approach.KeywordsA posteriori error analysisElliptic variational inequalitiesFinite elementsOptimal controlStationarity

Finite element exterior calculus (FEEC) has been developed as a systematic framework for constructing and analyzing stable and accurate numerical methods for partial differential equations by employing differential complexes. This paper is devoted to analyzing the convergence of adaptive mixed finite element methods for the Hodge Laplacian equations based on FEEC without considering harmonic forms. A residual type a posteriori error estimate is obtained by using the Hodge decomposition, the regular decomposition, and bounded commuting quasi-interpolants. An additional marking strategy is added to ensure the quasi-orthogonality, based on which the convergence of adaptive mixed finite element methods is obtained without assuming the initial mesh size is small enough.